Orientation Dependence and Exact Groups for Ridge-Routing Sandpiles

A fitting orientation of a simplicial rooted forest specifies an unsigned chip-firing rule on its nonroot ridges. We show that the resulting sandpile group can depend on that orientation: in every dimension d >= 2, a six-facet rooted forest with relative boundary determinant one admits cyclic groups of orders d^6-d^3-d and d^6-2d. In contrast, if each ridge belongs to at most two facets, the abstract group is independent of the fitting orientation. We give its cyclic decomposition and exact formulas for recurrent states, the identity, single-chip avalanches and individual addition periods. Even in this thin case, labeled addition periods can change. The proofs use an integral cycle-and-chain splitting and standard directed-sandpile theory. These results analyze a specified routing model; they do not settle the general AIM request for a higher-dimensional chip-firing model or establish a canonical torsor. The baseline model and known linear algebra are credited. AI-assisted, self-audited and unrefereed; novelty and absolute priority remain undetermined.